Algebraic Graph Theory Seminar - Brendan Rooney

Title: Efficient $(j,k)$-Domination

Abstract:

A function $f:V(G)\rightarrow\{0,\ldots,j\}$ is an efficient $(j,k)$-dominating function on $G$ if $\sum_{u\in N[v]}f(u)=k$ for all $v\in V(G)$ (here $N[v]=N(v)\cup\{v\}$ is the closed neighbourhood of $v$). Efficient $(j,k)$-domination was introduced by Rubalcaba and Slater (2007) as a generalization of perfect domination, and efficient $k$-domination. We look at efficient domination on regular graphs, applying some standard tools from linear algebra and algebraic graph theory. Using these ideas we give a partial characterization of the values $k$ for which the Hamming graphs $H(q,d)$ are efficiently $(1,k)$-dominatable.